Andrews / Berndt Ramanujan's Lost Notebook

Part II
1. Auflage 2009
ISBN: 978-0-387-77766-5
Verlag: Springer
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 420 Seiten

In the spring of 1976, George Andrews of Pennsylvania State University visited the library at Trinity College, Cambridge, to examine the papers of the late G.N. Watson. Among these papers, Andrews discovered a sheaf of 138 pages in the handwriting of Srinivasa Ramanujan. This manuscript was soon designated 'Ramanujan's lost notebook.' The 'lost notebook' contains considerable material on mock theta functions and so undoubtedly emanates from the last year of Ramanujan's life. It should be emphasized that the material on mock theta functions is perhaps Ramanujan's deepest work.

Autoren/Hrsg.

Andrews, George E.

Berndt, Bruce C.

Inhaltsverzeichnis

1;Preface;7
2;Contents;8
3;Introduction;12
4;The Heine Transformation;16
4.1;Introduction;16
4.2;Heine's Method;17
4.3;Ramanujan's Proof of the q-Gauss Summation Theorem;21
4.4;Corollaries of (1.2.1) and (1.2.5);25
4.5;Corollaries of (1.2.6) and (1.2.7);33
4.6;Corollaries of (1.2.8), (1.2.9), and (1.2.10);35
4.7;Corollaries of Section 1.2 and Auxiliary Results;38
5;The Sears--Thomae Transformation;55
5.1;Introduction;55
5.2;Direct Corollaries of (2.1.1) and (2.1.3);55
5.3;Extended Corollaries of (2.1.1) and (2.1.3);56
6;Bilateral Series;62
6.1;Introduction;62
6.2;Background;63
6.3;The 11 Identity;65
6.4;The 22 Identities;71
6.5;Identities Arising from the Quintuple Product Identity;77
6.6;Miscellaneous Bilateral Identities;83
7;Well-Poised Series;89
7.1;Introduction;89
7.2;Applications of (4.1.3);90
7.3;Applications of Bailey's Formulas;97
8;Bailey's Lemma and Theta Expansions;105
8.1;Introduction;105
8.2;The Main Lemma;105
8.3;Corollaries of (5.2.3);107
8.4;Corollaries of (5.2.4) and Related Results;115
9;Partial Theta Functions;121
9.1;Introduction;121
9.2;A General Identity;122
9.3;Consequences of Theorem 6.2.1;123
9.4;The function (a,q);137
9.5;Euler's Identity and Its Extensions;141
9.6;The Warnaar Theory;149
10;Special Identities;156
10.1;Introduction;156
10.2;Generalized Modular Relations;157
10.3;Extending Abel's Lemma;165
10.4;Innocents Abroad;172
11;Theta Function Identities;180
11.1;Introduction;180
11.2;Cubic Identities;182
11.3;Septic Identities;187
12;Ramanujan's Cubic Class Invariant;202
12.1;Introduction;202
12.2;n and the Modular j-Invariant;206
12.3;n and the Class Invariant Gn;210
12.4;n and Modular Equations;211
12.5;n and Modular Equations in the Theory of Signature 3;215
12.6;n and Kronecker's Limit Formula;221
12.7;The Remaining Five Values;224
12.8;Some Modular Functions of Level 72;225
12.9;Computations of n Using the Shimura Reciprocity Law;228
13;Miscellaneous Results on Elliptic Functions and Theta Functions;232
13.1;A Quasi-theta Product;232
13.2;An Equivalent Formulation of (10.1.1) in Terms of Hyperbolic Series;233
13.3;Further Remarks on Ramanujan's Quasi-theta Product;238
13.4;A Generalization of the Dedekind Eta Function;241
13.5;Two Entries on Page 346;245
13.6;A Continued Fraction;247
13.7;Class Invariants;248
14;Formulas for the Power Series Coefficients of Certain Quotients of Eisenstein Series;250
14.1;Introduction;250
14.2;The Key Theorem;254
14.3;The Coefficients of 1/Q(q);264
14.4;The Coefficients of Q(q)/R(q);280
14.5;The Coefficients of (P(q)/3)/R(q) and (P(q)/3)2/R(q);287
14.6;The Coefficients of (P(q)/23)/Q(q);291
14.7;Eight Identities for Eisenstein Series and Theta Functions;294
14.8;The Coefficients of 1/B(q);297
14.9;Formulas for the Coefficients of Further Eisenstein Series;305
14.10;The Coefficients of 1/B2(q);307
14.11;A Calculation from p;319
15;Letters from Matlock House;320
15.1;Introduction;320
15.2;A Lower Bound;321
15.3;An Upper Bound;329
16;Eisenstein Series and Modular Equations;333
16.1;Introduction;333
16.2;Preliminary Results;334
16.3;Quintic Identities: First Method;337
16.4;Quintic Identities: Second Method;344
16.5;Septic Identities;351
16.6;Septic Differential Equations;359
17;Series Representable in Terms of Eisenstein Series;361
17.1;Introduction;361
17.2;The Series T2k(q);362
17.3;The Series Un(q);368
18;Eisenstein Series and Approximations to ;371
18.1;Introduction;371
18.2;Eisenstein Series and the Modular j-Invariant;372
18.3;Eisenstein Series and Equations in : First Method;373
18.4;Eisenstein Series and Equations in : Second Method;376
18.5;Page 213;381
18.6;Ramanujan's Series for 1/;381
19;Miscellaneous Results on Eisenstein Series;391
19.1;A generalization of Eisenstein Series;391
19.2;Representations of Eisenstein Series in Terms of Elliptic Function Parameters;392
19.3;Values of Certain Eisenstein Series;393
19.4;Some Elementary Identities;394
20;Location Guide;397
21;Provenance;403
22;References;406
23;Index;420